Astronomy 161 All Homework Answers
Suppose that an organism replicates itself each second. If you start with a single specimen, what will the final population be after 10 seconds?
2^9 The final production after 10 seconds will be 512 specimen.
How was Earth's carbon dioxide atmosphere changed into today's oxygen-rich atmosphere? How long did that transformation take?
Approximately 2 billion years ago, the presence of cyanobacteria on the surface of waters on Earth contributed to the change in Earth's atmosphere. The cyanobacteria colonies photosynthesized by taking in the sunlight and abundant carbon dioxide in Earth's primitive atmosphere. All the while it consumed carbon dioxide it produced and expelled oxygen into the atmosphere. Initially this oxygen was absorbed and removed from the atmosphere by oxidation. But eventually everything that had been oxidized became unable to absorb any more of the oxygen being released into the atmosphere, thus it began to build up within the atmosphere. This process is suspected to have taken about 1,750,000,000 years ago since the current oxygen level has only been reached about 250 million years ago.
A sunspot appears only 70% as bright as the surrounding photosphere. The photosphere has a temperature of approximately 5780 K. What is the temperature of the sunspot?
Brightness (energy per second falling on a unit area of your detector) is directly proportional to flux (energy per second being radiated away from a unit area of the source). Flux, in turn, is proportional to the fourth power of the temperature (Stefan-Boltzmann Law). So, T4sunspot/T4photosphere = bsunspot/bphotosphere T4sunspot = T4photosphere (bsunspot/bphotosphere) Tsunspot = Tphotosphere (bsunspot/bphotosphere)1/4 = 5780 K (0.7)1/4 = 5780 x 0.915 = 5290 K
Archaeological samples are often dated by radiocarbon dating. The half-life of carbon-14 is 5,700 years. a) After how many half-lives will the sample have only 1/64 as much carbon-14 as it originally contained? b) How much time will have passed? c) If the daughter product of carbon-14 is present in the sample when it forms (even before any radioactive decay happens), you cannot assume that every daughter you see is the result of carbon-14 decay. If you did make this assumption, would you overestimate or underestimate the age of the sample?
During each half-life the radioactive sample will decrease by half. 1/64 = (1/2)6 so six half-lives have passed. (6 x 5,700) = 34,200 years have passed. By assuming that all daughter nuclei are from decays, you would be overestimating the number of decays which have taken place and thus overestimating the age of the sample.
Astronomers think that most planetary rings eventually dissipate. Explain why the rings do not last forever. Describe and explain a mechanism that keeps planetary rings from dissipating.
Each ring particle is in a Keplerian orbit, but it is also influenced by the gravitational pull of the planet's moons. This causes deviations in the orbits which can lead to collisions. These can change the energy of a particle, causing it to move to an orbit outside the ring. Shepherd satellites can help to keep particles in a ring. If a ring particle drifts outward from the ring, the shepherd satellite above it (moving more slowly) pulls it back, causing it to lose energy and return to the ring. If the particle drifts inward from the ring, the shepherd satellite below it (moving faster) pulls it forward, causing it to gain energy and return to the ring.
What is the advantage of launching satellites from spaceports located near the equator? Would you expect satellites to be launched to the east or to the west? Why?
Earth rotates toward the east, so to give the satellite the biggest "kick" possible it should be launched toward the east so that it begins with its launch velocity with respect to Earth PLUS the velocity of Earth's motion. And to pick up the most velocity from Earth, the launch should be made at the equator where Earth's rotation is fastest.
In the proton - proton chain, the mass of four protons is slightly greater than the mass of a helium nucleus. Explain what happens to this "lost" mass.
Einstein's famous equation E = mc2 is a statement that mass is a form of energy. In the proton - proton chain a small amount of mass disappears and is converted into a large amount of energy. This energy manifests itself as the kinetic energy of the particles created in the proton - proton chain: positrons, neutrinos, protons, deuterons, etc.
The Moon is on the meridian at your seaside home, but your tide calendar does not show that it is high tide. What might explain this apparent discrepancy?
Friction between the ocean basins and the oceans themselves cause Earth to drag the tidal bulges forward as it rotates beneath then. So, the Moon is about ten degrees behind a line drawn through the bulges. If the Moon is directly overhead, the high tide has already passed.
The inner and outer diameters of Saturn's B Ring are 184,000 and 235,000 km, respectively. Use this information to find the ratio of periods of particles at these two diameters. Does the B Ring orbit like a solid disk or like a collection of separate particles?
From Kepler's Third Law we know that the square of the period of a particle in gravitational orbit is proportional to the cube of the semi-major axis of its orbit. So Pinner/Pouter = 0.69. Thus the inner edge of the B Ring orbits faster (shorter period) than does the outer edge. Each semi-major axis distance within the ring orbits with a separate period, and the ring does not orbit like a solid body.
Suppose astronomers announce the discovery of a new planet around a star with a mass equal to the Sun's. The planet has an orbital period of 87 days. Is that planet in the habitable zone for a Sun-like star?
If a planet orbiting a Sun-like star that has a mass equal to our Sun has an orbital period of 87 days, it is not in the habitable zone. For a planet to be in the habitable zone of any star with a mass equal to the Sun's it must be far enough away from the Sun to have an orbital period ranging between that of approximately Venus's 224 days and Mar's 686 days. The planet with the orbital period of 87 days shares almost an identical orbit to Mercury's orbital period of 87.97 days. Thus, like Mercury, the planet is not in the habitable zone because it is too close to its star. A planet is too close to a star equal in mass to the Sun, it will be unable to sustain life due to its lack of water caused by the planet's proximity to the star. This close range would cause any water on the planet to boil and evaporate because of the amount of heat radiated from the star. This intense heat and lack of water make the planet inhabitable for any life.
A favorite object of amateur astronomers is the double star Albireo, with one of its components a golden yellow and the other a bright blue. What do these colors tell you about the relative temperatures of the two stars?
In order to appear yellow, the star's spectrum must peak in the low-wavelength part of the spectrum, so that an observer will see the red, orange, and yellow light mixed together, but not much of the blue and violet. In order to appear blue, the star's spectrum must include all the visible colors, but must include more blue than longer-wavelength colors. According to Wien's Displacement Law, the blue star, with a spectrum peaking in the short-wavelength part of the spectrum, will be hotter than the yellow star, which has a spectrum peaking at longer wavelengths.
Explain the process that drives volcanism on Jupiter's moon Io.
In order to be volcanically active, Io must have hot magma beneath its crust. The source of the heat to keep the interior molten is tidal flexing caused by Jupiter. Io's orbit is not perfectly circular because of the influence of the other Galilean moons. Thus its rate of rotation about its axis (perfectly regular) is not in step with the rate at which it moves along its elliptical orbit (which varies - Kepler's second law). The same side does not always face Jupiter, although Jupiter is tugging on it. This results in tidal flexing and friction.
The largest astronomical refractor has an aperture of 1 meter. List several reasons why it would be impractical to build a refractor with twice this aperture.
Increasing the diameter by a factor of two means an increase in volume and weight by a factor of eight. Large lenses sag and distort under their own weight, especially because, unlike mirrors, they can be supported only by the edge. Lenses have two surfaces, unlike mirrors which have only one. Thus they are more difficult and more expensive to manufacture. A thicker layer of glass means more scattering by impurities and more chromate aberration.
The asteroid Ida (mass 4.2x1016 kg) is attended by a tiny asteroidal moon, Dactyl, which orbits Ida at an average distance of 90 km. Neglecting the mass of the tiny moon, what is Dactyl's orbital period in hours?
Kepler's Third Law applies to this situation, but we need to use it in Newton's general formulation rather than Kepler's original formulation (which applies only to the planets around the Sun). P2 = 4p2 R3/(G MIda) = 4p2 (9x104 m)3/[(6.673x10-11 m3/(kg s2) (4.2x1016 kg)] = 1.03x1010 (seconds)2 P = 1.01x105 seconds P = (1.01x105 sec) (1 min/60 sec) (1 hour/60 min) = 28.2 hours
Titan contains abundant amounts of methane. What process destroys methane in this moon's atmosphere?
Photodissociation is the process by which ultraviolet radiation from the Sun breaks methane molecules apart, somewhat like the dissociation of oxygen and ozone in Earth's atmosphere. The resulting fragments recombine to form other hydrocarbons. The continued existence of large amounts of methane in the atmosphere suggests replenishment through volcanic outgassing.
Two of the three atoms in a molecule of water (H2O) are hydrogen. Why are Earth's oceans not fusing hydrogen into helium and setting Earth ablaze?
Protons (hydrogen nuclei) carry positive electric charge and thus repel each other. In order for two protons to come close enough to each other that the nuclear force (which is attractive but very short ranged) can bind them, they must be moving at very high speeds. This is possible at temperatures in the solar core, but it is very hard to achieve these temperatures on Earth. This is why fusion power has been so difficult to develop.
If you were on Mars, how often would you see retrograde motion of Earth in the martian night sky?
Retrograde motion is observed when one planet passes another in their orbits. The period of Mars in its orbit around the Sun is roughly two years. If Earth and Mars pass each other (and thus exhibit retrograde motion) on one side of the Sun, then one year later Earth will be on that same side of the Sun again, but Mars will be half way around on the far side of the Sun. After another year both planets will be together again at their original locations, and thus exhibit retrograde motion. Thus the time between consecutive retrograde motions is approximately two years.
Why do we find rocky material everywhere in the Solar System but large amounts of volatile material only in the outer regions?
Rocky materials are refractory which means they solidify at relatively high temperatures. In the early solar nebula these refractory materials could form solids everywhere, even in the hot region close to the protosun. As solids they could participate in the accretion process to form planets. Volatile materials solidify only at very cold temperatures, such as those in the outer regions of the solar nebula. They could participate in the accretion process only there. An additional reason for the separation of these materials is that the protosun went through an unstable episode known as its T-Tauri phase during which it erupted and blew the volatile material out of the inner part of the solar nebula.
Sirius is actually a binary pair of two A-type stars. The brighter of the two stars is called the "Dog Star" and the fainter is called the "Pup Star" because Sirius is in the constellation Canis Major (meaning "Big Dog"). The Dog Star appears about 6800 times brighter than the Pup Star, even though both stars are the same distance from Earth. Compare the temperatures, luminosities, and sizes of these two stars.
Temperatures: Since they are both A-type stars, they both have approximately the same temperature. Luminosities: In order to be so much brighter at the same distance, the Dog Star must be much more luminous. Since b = L/(4pd2), 6800 times brighter at the same d means 6800 times more luminous. Sizes: The flux (energy/sec/area) of a star depends only on temperature (F=sT4), thus both of these stars have about the same flux. In order to have a larger luminosity, the Dog Star must have 6800 times more area. And its radius must be SQRT(6800) = 82.4 times larger.
By what criteria did Pluto fail to be considered a planet under the new IAU definition? Explain how this decision demonstrates the self-correcting nature of science.
The IAU set three criteria which a body must meet before officially being considered a planet: 1) It orbits the Sun or another star. 2) It has enough mass that gravity has pulled it into a roughly spherical shape. 3) It has sufficient gravitational influence to clear all debris out of its orbit. Pluto meets the first two of these criteria, but not the last. The IAU definition of "planet" was changed to meet accumulating evidence that Pluto was unlike the other eight traditional planets and more like many other newly discovered bodies. Science relies on experimental results and must thus remain flexible to accommodate new evidence.
The apparent diameter of the Moon in the sky is approximately ½°. How long does it take the Moon to move 360°? About how long does it take the Moon to move a distance equal to its own diameter across the sky?
The Moon, like everything else, takes about 24 hours to move 360°. This is 360°/(24hr x 60 min/hr) = 0.25°/minute. So it takes the Moon about two minutes to move through half a degree.
What are the likely sources of Earth's water?
The accretion process which formed Earth included minerals which contained water. This water was released from the minerals in the hot interior of Earth and vented to the surface through volcanoes. For a long time it was thought that comets were an important source of Earth's water. However, we have recently been able to sample the water on several comets, and we have found that the isotopic ratio in the hydrogen (i.e. the ratio of deuterium to normal hydrogen) is significantly different from that in Earth's oceans. Asteroids are now considered to be a more important source for Earth's water.
Calculate the orbital radius of the Kirkwood gap that is in a 3:1 orbital resonance with Jupiter.
The asteroid belt is closer to the Sun than is Jupiter. So the asteroids have shorter orbital periods than Jupiter. An asteroid in a 3:1 orbital resonance with Jupiter is one which has an orbital period one third that of Jupiter, so the asteroid would make three complete orbits of the Sun while Jupiter makes one. The orbital period of Jupiter is 11.863 years (Appendix 4). So the asteroid in question would have a period of 3.954 years. The orbital radius of such an asteroid can be found from Kepler's Third Law.
If an alien astronomer observed a plot of the light curve as Jupiter passed in front of the Sun, by how much would the Sun's brightness drop during the transit?
The brightness of the Sun is proportional to the area which emits light. If Jupiter passes in front of the Sun, it will block the amount of the Sun's area equal to its own area. So the Sun's observed brightness would decrease by this fraction. Fractional decrease = Area of Jupiter/Area of Sun = π R2Jupiter/π R2Sun =(7.1492x104 km)2/(6.9627x105 km)2 = 0.0105 (about 1%)
The Sun has a radius equal to about 2.3 light-seconds. Explain why a gamma ray produced in the Sun's core does not emerge from the Sun's surface 2.3 seconds later
The distance 2.3 light-minutes is, by definition, the distance light travels in 2.3 minutes. If light traveled from the core to the surface in a straight unbroken line it would reach the surface in 2.3 minutes. But, it does not. Along the way it is absorbed by an atom, re-radiated in some random direction, and then absorbed again. This happens a very large number of times, and the actual time for emergence from the surface can be a hundred thousand years or more.
If photons of blue light have more energy that photons of red light, how can a beam of red light carry as much energy as a beam of blue light?
The energy in a beam of light depends on how many photons are in the beam as well as how much energy each photon carries. If there are more photons in the red beam than in the blue beam, the red beam can carry as much (or even more) energy as the blue beam.
What would happen to our ability to measure stellar parallax if we were on the planet Mars? What if we were on Venus or Jupiter?
The equation for parallax is based on this long thin triangle where d is the distance to the star, s is one AU, and a is the parallax angle. The relationship among these quantities is d = s/a. If a is measured in seconds of arc the resulting distance is measured in parsecs. If we were on Mars, s would be about 1.5 AU, and we would be able to measure parallax more easily. In fact, a star 1.0 parsecs away would have a parallax angle of 1.5 arcsecs rather than one arcsec if viewed from Earth. If we were on Venus, s = 0.72 AU, and parallax would be harder to measure. That same star, 1.0 parsecs distant, would have a parallax angle of 0.72 arcsec. On Jupiter s = 5.2 AU. That star 1.0 parsecs distant would have a parallax angle of 5.2 arcsec, much more easily measured.
Physicists describe certain properties, such as angular momentum and energy, as being conserved. What does this mean? Do conservation laws imply that an individual object can never lose or gain angular momentum or energy? Explain your reasoning
The fact that angular momentum is a conserved quantity says that the angular momentum of the whole universe is a constant and cannot be changed. But the angular momentum of an individual object can change - as long as the angular momentum of the rest of the universe changes in the opposite way so the total remains the same. In a practical sense, what we usually do is consider a small portion of the universe which is isolated so that the conserved quantity cannot be changed by interaction with anything outside. For example, I did a demonstration in lecture with a "frictionless" rotating stool. As I changed the mass configuration the rotational speed changed to conserve the angular momentum of this "isolated" system.
Explain why the Moon's core is cooler than Earth's.
The heat energy contained in a body is proportional to its volume, and this increases as the cube of its size (linear dimension, radius, etc). This heat escapes through the surface, but the surface area increases only as the square of its size. So larger bodies cool more slowly because they have more volume of heat per unit area of surface. The Moon is a smaller body than is Earth, so it cooled more rapidly. This is also the reason Moon rocks formed before the oldest rocks on Earth.
Why do scientists think that Mars and Venus were once more habitable but no longer are?
The key to life as we know it is liquid water. And there is abundant evidence that Mars once contained liquid water - large amounts of it - on its surface: sedimentary rocks, evaporates, "blueberries" (hematite concretions), mud splashes, etc. So, in the past, Mars was much more habitable than it is now. But, because of its distance from the Sun, dissolving the greenhouse gases in these oceans reduced the temperature below the freezing point. The water is now believed to exist as permafrost below the surface. The case for Venus is more marginal, but, since the Sun was dimmer in the past, some scientists suggest it may have been cool enough to have liquid water on its surface as well.Then the temperature rose, the greenhouse gases vaporized, and the present "runaway" greenhouse effect resulted.
If Kepler had lived on Mars, would he have deduced the same empirical laws for the motion of the planets? Explain.
The key to this problem is that Earth's place in the solar system is not unique, nor is that of Mars. No matter which planet you observe from, the laws for the motion of planets around the Sun (1. elliptic orbits, 2. equal areas in equal times, 3. P2 = a3) will be seen to be the same.
What are the primary reasons scientists have decided that the surfaces of Venus, Earth, and Mars are younger than those of Mercury and the Moon?
The main reason is that the surfaces of Mercury and the Moon are heavily cratered while those of Venus, Earth, and Mars are not. Most heavy cratering took place early in the solar system's history, and all these bodies were impacted. But the surfaces of Venus, Earth, and Mars have been reformed, mostly by volcanism, since that time. We can see volcanoes on Earth and Mars, and there are signs of volcanoes on Venus as well. There are no obvious volcanoes on Mercury or the Moon. When comparing Earth and the Moon, we actually have rock samples which have been dated to show Earth younger than the Moon.
If a 100 kg astronaut pushes on a 5000 kg satellite, and the satellite experiences an acceleration of 0.1 m/s2, what is the acceleration experienced by the astronaut in the opposite direction?
The net force experienced by the satellite is Fnet = Msat asat = (5000 kg) (0.1 m/s2) = 500 Newtons. By Newton's Third Law of Motion, the net force on the astronaut must also be 500 Newtons in the opposite direction. So the acceleration of the astronaut is aastronaut = Fnet/Mastronaut = 500 N/100 kg = 5.0 m/s2.
The Sun's visible "surface" is not a true surface, but a feature called the photosphere. Explain why the photosphere is not a true surface.
We usually think of a surface as something solid and impenetrable. The photosphere, however, is just the region between the convection zone - where the Sun's density is too great to allow radiation to travel very far - and the chromosphere - where the Sun is too rarified to produce much radiation. The photosphere isn't at a single radius, but rather it is a region about 500 km thick. As a result, when we look at the Sun we are looking at light which originated in this region, so it appears to be the visible "surface".
What evidence supports the theory suggesting that a mass extinction occurred as a consequence of an enormous impact on Earth 65 million years ago?
What evidence supports the theory suggesting that a mass extinction occurred as a consequence of an enormous impact on Earth 65 million years ago?
What is a protoplanetary disk? What are two reason that the inner part of the disk is hotter than the outer part?
When a large, roughly spherical mass of cold gas and dust collapses due to gravity, it rotates and flattens to form a disk. This is the protoplanetary disk, and it may form planets. The disk gains energy (and temperature) when material above and below the disk crashes into the disk. The material crashing into the inner part of the disk has fallen a longer distance than the material crashing into the outer part of the disk. This makes the inner part hotter than the outer part. Also, because of gravity, the greatest concentration of mass is at the center of the disk. This central mass becomes the protostar. As it collapses, it becomes hot and radiates this heat to the rest of the disk, more so in the inner part of the disk and less so in the outer part.
Your microwave oven cooks by vibrating water molecules at a frequency of 2.45 gigahertz (GHz), or 2.45x109 Hz. What is the wavelength, in centimeters, of the microwave's electromagnetic radiation?
c = f l Þ l = c/f = (2.99x108 m/s)/( 2.45x109 Hz) = 0.122 m = 12.2 cm
How long does Newton's cannonball, moving at 7.9 km/sec just above Earth's surface, take to complete one orbit around Earth?
speed = distance/time Þ time = distance/speed The distance around the Earth is its circumference, 2pRE , where RE = 6378 km from Appendix 4. Time = 2p (6378 km)/(7.9 km/s) = 5070 sec = 84 minutes = 1.41 hours
The Sun could get energy from gravitational contraction for a time period of G M2Sun / RSun L Sun. How long would the Sun last at its current luminosity? (Be careful with units!)
time = (6.67x10-11 Nm2/kg2)(1.99x1030 kg)2/(6.96x108 m)(3.839x1026 N m/s) = 9.88x1014 s (1 year/3.16x107 s)=3.14x107 yr=31.4 million years
